Abstract

During embryogenesis tissue layers undergo morphogenetic flow rearranging and folding into specific shapes. While developmental biology has identified key genes and local cellular processes, global coordination of tissue remodeling at the organ scale remains unclear. Here, we combine in toto light-sheet microscopy of the Drosophila embryo with quantitative analysis and physical modeling to relate cellular flow with the patterns of force generation during the gastrulation process. We find that the complex spatio-temporal flow pattern can be predicted from the measured meso-scale myosin density and anisotropy using a simple, effective viscous model of the tissue, achieving close to 90% accuracy with one time dependent and two constant parameters. Our analysis uncovers the importance of a) spatial modulation of myosin distribution on the scale of the embryo and b) the non-locality of its effect due to mechanical interaction of cells, demonstrating the need for the global perspective in the study of morphogenetic flow.

(a) Time course of Myosin visualized by sgh-GFP on apical surface, as described in . Root mean square contrast Crms for across the entire embryo surface is reported for each time point shown. Images have been normalized to range from 0 to 1 as described in the SI. (b) Histogram of average edge intensity. Dashed lines indicate minimum of distribution and 95th percentile determined once per dataset. Data shown are for time point 17 min (b’) As (b), except corrected for minimum and normalized to 95th percentile. (c) Histogram of edge intensities in dorsal and lateral regions, respectively.

(a) WT ensemble averaged flow field magnitude, averaged over embryo surface. Standard deviation is across samples. (b) Flow lines obtained by integrating flow field over time, for WT embryos −6 before to 3 min past CF formation. For simplicity of visualization and each line gets a random color assigned. (c) Cell area on the dorsal side of the embryo shrinks with time (measured using confocal microscope). indicates time starts from beginning of the experiment, at a time point during cellularization. (d–f) Time average length of flow lines for WT (d), twist, and (f) bcd nos tsl embryos, at times indicated in (b).

(a) Normalized signal strength of basal, apical, and polarized pools over time in the lateral ectoderm (outlined as dashed box in c). First gray shaded box at t < 0 min indicates times before CF formation (pre-CF), second shaded indicates GBE. (b) Automated extraction of polarization based on images, and quantitative summary as nematic tensor. Top left box shows cell outlines in part of a tissue, and a region of interest (ROI), that moves across the tissue. Bottom left box shows zoom on spatial signal in ROI. Colors indicate potentially different intensities of lines labeled i,j,k. Average intensity and length of lines in images are denoted I and L respectively. Radon transforms integrate signal along lines (cyan) of orientation α at normal-distance δ from the origin (purple). Bottom right inset shows sketch of resulting Radon-transformed signal. Note that lines are peaks at angle α, and distance δ, of height L*I after transformation. Top right inset shows definition of unit vector with orientation of edge i. Definition of local myosin tensor (only computed on apical surface, see ) for edge i is obtained by contracting unit edge vector with itself and weighted by line average intensity. (c) Magnitude of myosin anisotropy on pullback (see SI for definition). Dashed box indicates region of interest used to compute time traces in a. (d) Axis of myosin anisotropy (in cyan) overlayed on embryo labeled with his2Av-RFP in red, and eve-YFP in yellow. For simplicity of comparison, the field is only shown along even skipped stripes. For more detailed analysis see .

(a) Continuous representation of the myosin tensor. Black dot indicates coordinates on mesoscale and green cells represent the tissue. Edge i is highlighted in blue, and distance of edge i to a point x in the lattice is shown as a solid double-arrow. (b) Definition of myosin tensor on the mesoscale. Coordinates are indicated by x, edges detected by the radon transform are indicated by i. Detected edges contribute to a point by the normalized weight w(x,i), which we model as a suitably normalized Gaussian distribution. The resulting myosin tensor decomposes into a trace and a traceless part.

(a) Relative myosin anisotropy of basal pool, as computed by the magnitude of the traceless part over the total myosin tensor. (b) Relative myosin anisotropy of apical pool. (c) Time course of relative myosin anisotropy in the apical pool for representative time points. Definition and lookup table is the same as in (a,b). (d) Principal axes of anisotropic myosin tensor on apical surface, for times shown in (c). Major axis is shown in blue, minor in red. Magnitude is proportional to corresponding eigenvalue.

(a) Proposed mathematical description of the flow parameterizes complex mechanics of cytoskeleton in terms of the shear ν1 and ν2 bulk effective viscosities. The flow is driven by the force proportional to the divergence of the myosin tensor (see SI) on the right-hand-side of Equation 3a. Because effective viscosity tends to suppress velocity differences of neighboring cells, the response to local forcing is felt globally, e.g. effect of a local myosin perturbation results in local as well as non-local changes of the flow field. (b) Fit residual, comparing predicted flow field with measured flow field (see SI Finite Element implementation for a detailed definition of the residual) as a function of time. Both fields are normalized for average magnitude. The average magnitude of predicted velocity field defines one of our fitting parameters. Images of the single embryo are shown in – (c–e) Representative time points of morphogenetic flow: pre-CF (c), GBE (d) and VF (e). From top to bottom: spatial distribution of predicted (blue), measured (red) flow field, and residual (blue best agreement, red worst, on a scale from 0 to 1). For the case of VF flow, predictive model is modified to allow for a ‘cut’ in ventral region (see SI text, and for detail).

(a) Schematic of the finite element model including the ‘cut’ used to model ventral furrow formation. Shown is a triangulation of the embryo from a ventral perspective, anterior to the left. The ‘cut’ is realized by removing all edges within the red outlined box and introducing a contractile force(shown in green) normal to the edge as boundary condition.(b) Time dependence of B, representing ratio of viscosities (see for detail). (c) Model summary, with explanation of temporal dependence of model parameters. Ten percent confidence intervals are indicated in brackets.

(a) Fit residual as in , for twi, and bcd nos tsl mutants (7, and 7 embryos in ensemble). WT is shown as reference. (b) Amplitude of basal myosin pool along DV axis for WT and mutants in (a). (c) Polarized apical myosin in mutants shown in (a) as function of time. (d) Theoretical comparison of DV constant basal pool (i.e. no gradient in DV direction) (left column), or DV constant anisotropic apical pool (i.e. no gradient in DV direction) (right column) with predicted flow based on full myosin tensor (compare to respectively). Black arrows indicate flow field topology, and red dots the fixed point from prediction based off of full myosin tensor. Model parameters are the same as previously determined for the WT ().